Higher order elicitability Johanna F. Ziegel University of Bern joint work with Fernando Fasciati, Tobias Fissler, Tilmann Gneiting, Alexander Jordan, Fabian Kr¨ uger and Natalia Nolde 2 June 2017 Van Dantzig Seminar 1 / 52
Outline 1. Elicitability ◮ Definition and a simple example ◮ Risk measures ◮ k -Elicitability ◮ Osband’s principle 2. Evaluating forecasts of expected shortfall ◮ Absolute forecast evaluation ◮ Classical comparative forecast evaluation ◮ Comparative forecast evaluation with Murphy diagrams 3. Summary 2 / 52
Outline 1. Elicitability ◮ Definition and a simple example ◮ Risk measures in banking ◮ k -Elicitability 2. Evaluating forecasts of expected shortfall ◮ Absolute forecast evaluation ◮ Classical comparative forecast evaluation ◮ Comparative forecast evaluation with Murphy diagrams 3. Summary 3 / 52
Elicitability Let P be a class of probability measures on O ⊆ R d . Let T : P → A , F �→ T ( F ) be a functional where A ⊆ R . Definition A scoring (or loss) function S : A × O → R is consistent for T relative to P , if E F S ( T ( F ) , Y ) ≤ E F S ( x , Y ) , F ∈ P , x ∈ A . It is strictly consistent if “=” implies x = T ( F ). The functional T is called elicitable relative to P if there exists a scoring function S that is strictly consistent for it. In other words T ( F ) = arg min x ∈ A E F S ( x , Y ) . 4 / 52
A simple example – the mean Let Y be a random variable with distribution function F . Suppose that E F Y 2 < ∞ . Then, x ∈ R E F ( Y − x ) 2 . E F Y = arg min ◮ The mean is elicitable with respect to the class of all probability measures with finite second moment. 5 / 52
A simple example – the mean Theorem (Savage, 1971) Let P be a class of probability measures with finite first moments. Let φ be a (strictly) convex function such that E F φ ( Y ) exists and is finite for all F ∈ P . Then, S ( x , y ) = φ ( y ) − φ ( x ) − φ ′ ( x )( y − x ) is (strictly) consistent for the mean. ◮ Under suitable assumptions on P , the Bregman functions are the only consistent scoring functions for the mean. ◮ Choosing φ ( y ) = y 2 / (1 + | y | ) shows that the mean is elicitable with respect to the class of all probability measures with finite first moment. 6 / 52
A simple example – the mean Theorem (Savage, 1971) Let P be a class of probability measures with finite first moments. Let φ be a (strictly) convex function such that E F φ ( Y ) exists and is finite for all F ∈ P . Then, S ( x , y ) = φ ( y ) − φ ( x ) − φ ′ ( x )( y − x ) is (strictly) consistent for the mean. ◮ Under suitable assumptions on P , the Bregman functions are the only consistent scoring functions for the mean. ◮ Choosing φ ( y ) = y 2 / (1 + | y | ) shows that the mean is elicitable with respect to the class of all probability measures with finite first moment. 6 / 52
A simple example – the mean Theorem (Savage, 1971) Let P be a class of probability measures with finite first moments. Let φ be a (strictly) convex function such that E F φ ( Y ) exists and is finite for all F ∈ P . Then, S ( x , y ) = φ ( y ) − φ ( x ) − φ ′ ( x )( y − x ) is (strictly) consistent for the mean. ◮ Under suitable assumptions on P , the Bregman functions are the only consistent scoring functions for the mean. ◮ Choosing φ ( y ) = y 2 / (1 + | y | ) shows that the mean is elicitable with respect to the class of all probability measures with finite first moment. 6 / 52
Why is elicitability interesting? Generalized regression/M-estimation Assume the following model T ( L ( Y | Z )) = m ( Z , β ) parametrized by β ∈ Θ and let S be a strictly consistent scoring function for T . Suppose we have iid observations ( z i , y i ), i = 1 , . . . , n from ( Z , Y ). Then, we can estimate β by n 1 ˆ � β = arg min S ( y i , m ( z i , β ′ )) . n β ′ ∈ Θ i =1 ◮ Least squares regression ◮ Quantile regression ◮ Logistic regression 7 / 52
Why is elicitability interesting? Generalized regression/M-estimation Assume the following model T ( L ( Y | Z )) = m ( Z , β ) parametrized by β ∈ Θ and let S be a strictly consistent scoring function for T . Suppose we have iid observations ( z i , y i ), i = 1 , . . . , n from ( Z , Y ). Then, we can estimate β by n 1 ˆ � β = arg min S ( y i , m ( z i , β ′ )) . n β ′ ∈ Θ i =1 ◮ Least squares regression ◮ Quantile regression ◮ Logistic regression 7 / 52
Why is elicitability interesting? Forecast comparison/Model selection Suppose we have sequences of competing forecasts x A 1 , . . . , x A n , x B 1 , . . . , x B n for T and observations y 1 , . . . , y n . Let S be a strictly consistent scoring function for T . Then it is natural to prefer method A over method B if n n 1 i , y i ) < 1 � � S ( x A S ( x B i , y i ) . n n i =1 i =1 8 / 52
Why is elicitability interesting? Forecast comparison/Model selection Suppose we have sequences of competing forecasts x A 1 , . . . , x A n , x B 1 , . . . , x B n for T and observations y 1 , . . . , y n . Let S be a strictly consistent scoring function for T . Then it is natural to prefer method A over method B if n n 1 i , y i ) < 1 � � S ( x A S ( x B i , y i ) . n n i =1 i =1 8 / 52
Risk measures Let Y ∼ F be the single-period return of some financial asset. ◮ A risk measure assigns a real number to Y (interpreted as the risk of the asset). Risk measures are used for ◮ external regulatory capital calculation ◮ management, optimization and decision making ◮ performance analysis ◮ capital allocation 9 / 52
Risk measures Let Y ∼ F be the single-period return of some financial asset. ◮ A risk measure assigns a real number to Y (interpreted as the risk of the asset). Risk measures are used for ◮ external regulatory capital calculation ◮ management, optimization and decision making ◮ performance analysis ◮ capital allocation 9 / 52
Value at Risk and expected shortfall Let Y ∼ F , α ∈ (0 , 1). Value at Risk (VaR) VaR α ( Y ) = q α ( F ) = inf { x ∈ R : P ( Y ≤ x ) ≥ α } , Expected shortfall (ES) � α ES α ( Y ) = 1 VaR u ( Y ) d u . α 0 ◮ Profits are positive. ◮ We consider α close to zero ( α = 0 . 01, α = 0 . 025). ◮ Risky positions yield large negative values of VaR α and ES α . 10 / 52
Criticism of VaR as a risk measure Lack of super-additivity ◮ Usually there are several Y (1) , Y (2) , . . . to be considered with limited knowledge of their dependence. ◮ Goal: Bound on risk of the total � i Y ( i ) . ◮ VaR is not super-additive: There are Y (1) , Y (2) such that VaR α ( Y (1) + Y (2) ) < VaR α ( Y (1) ) + VaR α ( Y (2) ) . ◮ Problematic for risk aggregation. ◮ Counterintuitive to diversification. It is just a quantile. . . ◮ VaR α does not take sizes of losses beyond the threshold α into account. 11 / 52
Criticism of VaR as a risk measure Lack of super-additivity ◮ Usually there are several Y (1) , Y (2) , . . . to be considered with limited knowledge of their dependence. ◮ Goal: Bound on risk of the total � i Y ( i ) . ◮ VaR is not super-additive: There are Y (1) , Y (2) such that VaR α ( Y (1) + Y (2) ) < VaR α ( Y (1) ) + VaR α ( Y (2) ) . ◮ Problematic for risk aggregation. ◮ Counterintuitive to diversification. It is just a quantile. . . ◮ VaR α does not take sizes of losses beyond the threshold α into account. 11 / 52
Expected shortfall For continuous distributions, we have ES α ( Y ) = E ( Y | Y ≤ VaR α ( Y )) . ◮ ES α is a coherent risk measure, so in particular super-additive. ◮ It takes the entire tail of the distribution into account. ◮ Largest coherent risk measure is dominated by VaR α . ◮ It has a natural interpretation. 12 / 52
Elicitable and non-elicitable functionals Elicitable ◮ Mean, moments ◮ Median, quantiles/VaR ◮ Expectiles (Newey and Powell, 1987) Not elicitable ◮ Variance ◮ Expected Shortfall (Weber, 2006, Gneiting, 2011) Elicitable. . . ◮ coherent risk measures: Expectiles ◮ convex risk measures: Shortfall risk measures ◮ distortion risk measures: VaR and mean (Weber, 2006, Z 2014, Bellini and Bignozzi, 2014, Delbaen et al. 2015, Kou and Peng 2014, Wang and Z 2015) 13 / 52
k -Elicitability Let P be a class of probability measures on O ⊆ R d . Let T : P → A , F �→ T ( F ) be a functional where A ⊆ R k . Definition A scoring (or loss) function S : A × O → R is P -consistent for T , if E F S ( T ( F ) , Y ) ≤ E F S ( x , Y ) , F ∈ P , x ∈ A . It is strictly P -consistent if “=” implies x = T ( F ). The functional T is called k - elicitable relative to P if there exists a scoring function S that is strictly consistent for it. In other words T ( F ) = arg min x ∈ A E F S ( x , Y ) . 14 / 52
Elicitable functionals 1-Elicitable ◮ Mean, moments ◮ Median, quantiles/VaR ◮ Expectiles (Newey and Powell, 1987) 2-Elicitable ◮ Mean and variance ◮ Second moment and variance ◮ VaR and expected shortfall (Acerbi and Szekely, 2014, Fissler and Z, 2016) k -Elicitable ◮ Some spectral risk measures together with several VaRs at certain levels (Fissler and Z, 2016) 15 / 52
Recommend
More recommend
